A note on heroin epidemics

Mulone, G.; Straughan, Brian

doi:10.1016/j.mbs.2009.01.006

Cited by 108 publications

(69 citation statements)

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“…Besides incorporation of a saturated treatment function, our model also included an extra class of individuals, namely, those who have been successfully treated from heroin using. This class has been neglected in previous heroin epidemic models [6][7][8][9]. Much of our work will be focused on exploring the conditions for global stability of the heroin model with treatment.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a Heroin Epidemic Model with Saturated Treatment Function

Wangari¹,

Stone

2017

Journal of Applied Mathematics

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A mathematical model is developed that examines how heroin addiction spreads in society. The model is formulated to take into account the treatment of heroin users by incorporating a realistic functional form that "saturates" representing the limited availability of treatment. Bifurcation analysis reveals that the model has an intrinsic backward bifurcation whenever the saturation parameter is larger than a fixed threshold. We are particularly interested in studying the model's global stability. In the absence of backward bifurcations, Lyapunov functions can often be found and used to prove global stability. However, in the presence of backward bifurcations, such Lyapunov functions may not exist or may be difficult to construct. We make use of the geometric approach to global stability to derive a condition that ensures that the system is globally asymptotically stable. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users are mechanisms fuelling heroin epidemic proliferation.

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Section: Introductionmentioning

confidence: 99%

Analysis of a Heroin Epidemic Model with Saturated Treatment Function

Wangari¹,

Stone

2017

Journal of Applied Mathematics

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“…Due to the recent development of qualitative and quantitative analysis for disease transmission models, mathematical models have widely been applied to investigate the spread of habituation of getting addicted to drugs such as heroin (see e.g., [7,12,17,22,28,34,35]). …”

Section: Introductionmentioning

confidence: 99%

“…All the parameters are assumed to be positive. In addition to stability analysis for a drug-free equilibrium [35,Section 3], the stability analysis for a unique endemic equilibrium is achieved by Mulone and Straughan [22] when κ < β. Later, Wang et al [34] has formulated the model incorporating the mass action incidence rate and established the global stability of the drug-free equilibrium and the unique endemic equilibrium by means of the second compound matrix and under some conditions.…”

Section: Introductionmentioning

confidence: 99%

Global analysis of a multi-group SIR epidemic model with nonlinear incidence rates and distributed moving delays between patches

Muroya¹,

Kuniya²,

Enatsu³

2016

Proceedings of the 10'th Colloquium on the Qualitative Theory of Differential Equations (July 1–4, 2015, Szeged, Hungary) Edite

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Abstract. In this paper, applying Lyapunov functional approach, we establish sufficient conditions under which each equilibrium is globally asymptotically stable for a class of multi-group SIR epidemic models. The incidence rate is given by nonlinear incidence rates and distributed delays incorporating not only an exchange of individuals between patches through migration but also cross patch infection between different groups. We show that nonlinear incidence rates and distributed delays have no influence on the global stability, but patch structure has. Moreover, the present results generalize known results on the global stability of a heroin model with two delays considered in the recent literatures. We also offer new techniques to prove the boundedness of the solutions, the existence of the endemic equilibrium and permanence of the model.

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“…Behaviours which can result in adverse health effects have been represented, such as drinking [36,3,26,28,37,30], smoking [38], drug use [44,29], obesity [18,15] and eating disorders [14]. Even though the models for each social problem may appear mathematically similar at the onset, there are fundamental differences which must be catered for.…”

Section: Introductionmentioning

confidence: 99%

Modelling alcohol problems: total recovery

2012

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. (2013) 'Modelling alcohol problems : total recovery.', Ricerche di matematica., 62 (1). pp. 33-53. Further information on publisher's website:http://dx.doi.org/10.1007/s11587-012-0138-0Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/s11587-012-0138-0.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. 1We find that the model has two equilibrium points: one without alcohol problems and one where alcohol problems are endemic in the population. We compare our results with those of an existing model that does not allow for total recovery. We show that without total recovery, the threshold for alcohol problems to become endemic in the population is lowered. The endemic equilibrium solution is also affected, with an increased proportion of the population in the treatment class and a decreased proportion in the susceptible class.Including totally recovery does not determine whether the proportion of individuals with alcohol problems increases or decreases, however it does effect the size of the change. Parameter estimates are made from information regarding binge drinking where we find an increase in the recovery rate decreases the proportion of binge drinkers in the population.

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A note on heroin epidemics

Cited by 108 publications

References 6 publications

Analysis of a Heroin Epidemic Model with Saturated Treatment Function

Analysis of a Heroin Epidemic Model with Saturated Treatment Function

Global analysis of a multi-group SIR epidemic model with nonlinear incidence rates and distributed moving delays between patches

Modelling alcohol problems: total recovery

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